The Gordian complex of virtual knots by forbidden moves (Q2850720)

The \textit{Gordian distance} between two knots is the minimum number of crossing changes to obtain one knot from the other. Obviously the minimum is taken over all diagrams of the knots.NEWLINENEWLINE The Gordian complex of knots, introduced in [\textit{M. Hirasawa} and \textit{Y. Uchida}, J. Knot Theory Ramifications 11, No. 3, 363--368 (2002; Zbl 1004.57008)] is a simplicial complex whose vertices are knot types in the three sphere and whose faces correspond to sets of knots such that each pair of elements has Gordian distance one.NEWLINENEWLINE It is known that crossing change is not an unknotting operation on virtual knot diagrams. Forbidden moves were shown to be unknotting operations in [\textit{S. Nelson}, ibid. 10, No. 6, 931--935 (2001; Zbl 0997.57016)] and [\textit{T. Kanenobu}, ibid. 10, No. 1, 89--96 (2001; Zbl 0997.57015)].NEWLINENEWLINE In previous work, the authors of the paper under review together with \textit{K. Komura} and \textit{M. Shimozawa} [ibid. 21, No. 14, Paper No. 1250122, 11 p. (2012; Zbl 1275.57010)] defined the Gordian complex for virtual knots using an unknotting operation called the \(v\)-move.NEWLINENEWLINEIn the paper under review, the authors extend the notion of the Gordian complex to the case of virtual knots by using the forbidden moves and generalized Reidemeister moves. They show that for any virtual knot \(K_0\) and for any natural number \(n\), there exists a family of virtual knots \(K_1, \dots ,K_n\) such that for any pair of distinct knots in the family the virtual Gordian distance is one.